Data-Driven Computational Mechanics is a new computing philosophy that enables the evolution from conventional data-free methods to modern data-rich approaches. Its underlying concept relies on the reformulation of classical boundary value problems of elasticity and inelasticity such that material models, which are calibrated from experiments, are replaced by some form of experimental material data. On the one hand, Data-Driven Computational Mechanics eliminates some modeling errors and the associated uncertainty by employing experimental data directly. On the other hand, new sources of error emerge that are associated with the measurements and with the particular measuring technology employed. At least for the moment, there is no consensus regarding which sources of error are the most severe and therefore, there is still a long way to go.
Among recent developments, two principal approaches of Data-Driven Computational Mechanics can be distinguished. On the one hand, there is a direct one [Kirchdoerfer2016, Kirchdoerfer2017], whose methods are based on a discrete-continuous optimization problem that attempts to assign to each material point a point in the phase space that under fulfillment of the compatibility and equilibrium constraints is closest to the data set provided. Within that framework, some very interesting advances are reported, for instance, regarding nonlinear elasticity [Nguyen2018], general elasticity [Conti2018], inelasticity [Eggersmann2019], and mixed-integer quadratic optimization problems [Kanno2019]. On the other hand, there is an inverse one [Ibanez2017, Ibanez2018a, Ibanez2019], whose methods rely on an inverse approach that attempts to reconstruct from the data sets provided a constitutive manifold with a well-defined functional structure. In the context of these two families of methods, our recent work on approximate nonlinear optimization problems [Gebhardt2019d] represents a hybrid approach, targeting at a synergistic compromise that combines their strengths and mitigates some of their main weaknesses, in particular the high computational cost associated with the resolution of a discrete-continuous nonlinear optimization problem for the direct approach and the limitation to a special functional structure that only allows the explicit definition of stresses for the inverse approach.
Data-Driven Computational Dynamics, the application of Data-Driven Computational Mechanics principles to structural dynamics problems, is currently less developed, with a few papers published so far. For instance, in [Kirchdoerfer2018], the data-driven solvers for quasistatic problems developed in [Kirchdoerfer2016, Kirchdoerfer2017] were extended to dynamics, relying on variational time-stepping schemes such as the Newmark algorithm. In [Gonzalez2019], a thermodynamically consistent approach that relies on the “General Equation for Non-Equilibrium Reversible-Irreversible Coupling” formalism was presented, which enforces by design the conservation of energy and positive production of entropy.
The central goal of the present work is the formulation of an approximate nonlinear optimization problem for Data-Driven Computational Dynamics, which can be understood as the structural dynamics counterpart of the formulation previously developed by the authors in [Gebhardt2019d]. The proposed approximate nonlinear optimization problem relies on a discrete balance equation, which is inspired by a class of variational integrators [Marsden2001, Lew2003, Lew2004, Kale2007, Fong2008, Leyendecker2008, Betsch2010, Lew2016] and represents the dynamic equilibrium. Since in our approach, no special functional structure of the constitutive manifold is assumed, the existence of an energy function is in general not guaranteed, and therefore, energy-momentum methods [Gonzalez1996, McLachlan1999, Betsch2010, Romero2012, Gebhardt2019c]
are not directly applicable. Firstly, the proposed framework improves computational efficiency and robustness with respect to the type of solvers that rely on discrete-continuous optimization problems. In particular, our approximate nonlinear optimization problem can be solved with local Sequential Quadratic Programming methods, circumventing the necessity of employing meta-heuristic methods. Secondly, the proposed framework can deal with implicitly defined stress-strain relations and kinematic constraints, thus enlarging its range of applicability. Lastly, we consider the case of a geometrically exact beam element to demonstrate the advantages of our approach. Such a finite element model makes full use of our computational machinery. Be aware that our primary goal is a proof of concept for our new approximate nonlinear optimization approach for Data-Driven Computational Dynamics and therefore, the identification of the underlying constitutive manifold is not addressed here.
The remainder of this work is organized as follows: Section 2, the core of this article, presents two optimization problems for Data-Driven Computational Dynamics that are built upon a time integration approach inspired by a class of structure-preserving methods. The first one is an “exact” discrete-continuous nonlinear optimization problem that works directly with data sets. Such a problem can be considered as the starting point and is not going to be solved within this work. The second one is an “approximate” nonlinear optimization problem that relies on a general approximation of the underlying constitutive manifold, which circumvents completely the necessity of online handling of data sets. For both problems, we define the associated Lagrangian functions and derive explicitly the first order optimality conditions as well as the corresponding KKT matrices. In section 3, we specialize the proposed methodology for the geometrically exact beam finite element in a purely dynamic setting. This particular structural model has been chosen because it makes use of all elements of our approach. Section 4 presents simulation results that illustrate the capability of the derived approach with special emphasis on preserved quantities along the discrete motion, which is seen as the solution of a sequence of successive nonlinear optimization problems. Finally, in Section 5, we draw concluding remarks and propose future work.
2 Nonlinear optimization problems
The definition of successive nonlinear optimization problems in Data-Driven Computational Mechanics implies the partitioning of the considered time interval into subintervals such that . We consider an equidistant partitioning by a fixed time step, i.e., . A very simple scalar cost function to be minimized at time instant can be defined as
where the pair denotes continuous strain and stress variables and , respectively, a given finite data set contains strain and stress measurements , is a symmetric positive-definite weight matrix with inverse , and and are norms derived from an inner product. At this point, there is no necessity to specify since it depends on the structural model considered, which for now remains unspecified. The cost function (1) has to be minimized under the following constraints: i) the compatibility equation that enforces the equivalence between strain variables and displacement-based strains at time instant ,
is the vector of generalized coordinates andstands for the configuration manifold; ii) the discrete balance equation that establishes the dynamic equilibrium, for instance, we chose an approximation inspired by a family of variational integrators [Marsden2001, Lew2003, Lew2004, Kale2007, Fong2008, Leyendecker2008, Betsch2010, Lew2016] that renders the dynamic equilibrium at time instant as
in which represents the constant mass matrix,
is the Jacobian matrix of the displacement-based strains, similarly for , is the Jacobian matrix of the kinematic constraints at time instant , is the corresponding vector of Lagrange multipliers, and represents the vector of generalized external loads, similarly for ; and, iii) the kinematic constraints at time instant ,
a finite set of integrable restrictions that belongs to . As usual in the finite element setting, we assume that the Jacobian matrix of the displacement-based strains and the Jacobian matrix of the constraints are linear in , yielding substantial simplifications when calculating higher-order derivatives. Since the existence of an energy function is in general not guaranteed, energy-momentum methods are not directly applicable [Gonzalez1996, McLachlan1999, Betsch2010, Romero2012, Gebhardt2019c].
Now, to briefly investigate the conservation properties of the adopted time integration scheme, let us first neglect the external forces and define the discrete momenta:
These definitions are inspired by the discrete Legendre transforms that are widely used in the context of variational integrators. Having at hand the discrete momenta, the discrete balance equation can be rewritten as , which leads to the existence of a unique momentum at time instant . In this discrete setting, there are two possible definitions of linear momentum, namely
in which filters out all non-translational contributions and denotes the number of nodes. Provided that the system under consideration is invariant under translations, i.e., the orthogonality between the internal forces and the infinitesimal generator of translation is given, a unique discrete linear momentum does exist, namely , and is an invariant of the discrete motion, whose conservation law reads
Likewise, there are two possible definitions of angular momentum, namely
Similarly, provided that the system under consideration is invariant under rotations, i.e., the orthogonality between the internal forces and the infinitesimal generator of the rotation is given, a unique discrete angular momentum does exist, namely , and is an invariant of the discrete motion, whose conservation law reads
Lastly, to avoid problems caused by overdetermination and singular KKT matrices in the subsequent optimization problems, we eliminate the Lagrange multipliers from (3) by means of the null-space method. This requires a null-space basis matrix for with
, the system’s number of degrees of freedom, and, such that
Then, dynamic equilibrium adopts the form
where only the dependency on unknown quantities is explicitly indicated in .
2.1 The “exact” discrete-continuous nonlinear optimization problem
Definition 2.1 (Exact DCNLP).
Employing directly the strain and stress measurements, each successive “Data-Driven Computational Dynamics” problem can be defined as a discrete-continuous nonlinear optimization problem that can be stated as
Notice that the discrete variables at time instant appear only in the cost function. For fixed , the exact DCNLP becomes a smooth nonlinear optimization problem (NLP), referred to as . Any solution provides a set of values that (locally) minimizes the cost function for the fixed data point under the constraints given above.
The first-order optimality conditions of are:
where we define
The Lagrangian function of is
where are Lagrange multipliers of the compatibility equation at time instant , are Lagrange multipliers of the balance equation premultiplied by the null-space basis matrix evaluated at time instant , and are Lagrange multipliers of kinematic constraints at the instant . To derive the corresponding first-order optimality conditions, we calculate the variation of as
with the primal-dual NLP variable vector
Setting this to zero for any choice of the varied quantities yields the KKT conditions (14). ∎
Notice that and can be eliminated by substitution, but as we are interested in the problem’s global format, we are not going to eliminate anything unless strictly necessary.
The linearization of the variation of reads
where the KKT matrix is symmetric indefinite and can be written
for any constant vector . As the KKT matrix is non-singular, all local minima are strict minima and can be solved by local Sequential Quadratic Programming methods.
The overall DCNLP can be treated by meta-heuristic methods. Since it has no useful structure with respect to the discrete variables , a mathematically rigorous solution requires enumeration, that is, finding the minimal value over all measurements by solving every globally. Therefore we suggest a different approach: we propose to add suitable structure that enables us to replace the DCNLP with a single approximating NLP, as already done in the static case [Gebhardt2019d].
2.2 The “approximate” nonlinear optimization problem
The idea here is to replace the measurement data set by enforcing the state to belong to a reconstructed constitutive manifold that has a precise mathematical structure and that is derived from the data set. The underlying assumption is, of course, that such a constitutive manifold exists and that we can reconstruct a (smooth) implicit representation . The reconstructed constitutive manifold (an approximation) will enormously facilitate the task of the data-driven solver, avoiding the cost of solving a DCNLP, either by enumeration, or by heuristic or meta-heuristic methods which can in general only provide approximate solutions that strongly depend on the initial guess and whose convergence properties are inferior when compared to gradient-based methods.
An “approximate” constitutive manifold is defined as
for some accuracy . Additionally, physical consistency requires that implies and implies .
A constitutive manifold is said to be thermomechanically consistent if it is derived from an energy function such that the following functional structure holds [Crespo2017, Ibanez2017, Gebhardt2019d]:
However, in the case of new composite materials or metamaterials that exhibit non-convex responses, the reconstruction of the energy function may not be very convenient. More importantly, in some cases the formulation of an energy function may not even be possible. Thus, we adopt the constitutive manifold as introduced previously without assuming any special functional structure of the constitutive constraint . Further specializations are possible and should be instantiated for specific applications of the proposed formulation.
Definition 2.3 (Approximate NLP).
Each successive “Data-Driven Computational Dynamics” problem can be approximated as a nonlinear optimization problem of the form
The first-order optimality conditions of the approximate NLP are:
The Lagrangian function of the approximate NLP is given by
where are Lagrange multipliers that correspond to the enforcement of the strain and stress states to remain on the constitutive manifold.
To find the first-order optimality conditions, the variation of is calculated as
Setting this to zero for any choice of the varied quantities yields the KKT conditions (28). ∎
The linearization of the variation of can be expressed as
Here the KKT matrix can be written as